Tecnológico de Orizaba
4.2 Fundación del Centro Tecnológico de Orizaba
utilities. Decision trees with possibilistic likely dominance will be detailed in Section 4.4 and those with order of magnitude expected utility in Section 4.5. Possibilistic decision trees with Choquet integrals will be developed in Section 4.6 and polynomial cases of possibilistic Choquet integrals will be presented in Section 4.7.
Principle results of this chapter are published in [9].
4.2
Possibilistic decision trees
Possibilistic decision trees have the same graphical component as probabilistic ones (see Section 3.2 in Chapter 3) i.e. it is composed of a set of nodes N and a set of edges E. Like probabilistic decision trees, the set of nodes N contains three kinds of nodes i.e. N = D ∪ C ∪ LN where D is the set of decision nodes, C is the set of chance nodes and LN is the set of leaves. This is not the case of the numerical component which relies in the possibilistic framework:
• Arcs issuing from chance nodes are quantified by possibility degrees in the context of their parents. Formally, for any Ci ∈ C, the uncertainty pertaining to the more or less possible outcomes of each Ci is represented by a conditional possibility distribution πi on Succ(Ci), such that ∀N ∈ Succ(Ci), πi(N ) = Π(N |path(Ci)). To each node Ci∈ C, a possibilistic lottery LCi is associated relative to its outcomes.
• Then, a utility is assigned to each leaf nodes which can be numerical (e.g. currency gain) or ordinal (e.g. satisfaction) according to the decision criterion.
Example 4.1 The decision tree of Figure 4.1 is defined by
D = {D0, D1, D2}, C = {C1, C2, C3, C4, C5, C6} and LN = U = {0, 1, 2, 3, 4, 5}. Corre- sponding possibilistic lotteries to chance nodes are LC1 = h1/LD1, 0.5/LD2i, LC2 = h1/1, 0.7/2i,
LC3 = h1/1, 0/5i, LC4 = h0.2/0, 1/4i, LC5 = h1/1, 0.3/4i and LC6 = h1/2, 0.5/5i.
D0 D2 D1 C1 C2 C3 C4 C5 C6 1 5 0 4 1 4 2 5 1 2 1 0 0.2 1 1 0.3 1 0.5 1 0.7 1 0.5
Figure 4.1: Example of possibilistic decision tree
As we have seen in the previous chapter, solving decision trees amounts at building an optimal strategy δ∗ in ∆ (the set of sound and complete strategies).
Like in probabilistic decision trees, strategies can be evaluated and compared thanks to the notion of possibilistic lottery reduction: each chance node can be seen as a simple lottery (for the most right chance nodes) or as a compound lottery (for the inner chance nodes). Each strategy is thus a compound lottery and can be reduced to an equivalent simple one. Formally, the composition of possibilistic lotteries will be applied from the leafs of the strategy to its root, according to the following recursive definition for any Ni in N :
L(Ni, δ) = L(δ(Ni), δ) if Ni ∈ D Reduction(hπi(Xj)/L(Xj, δ)Xj∈Succ(Ni)i) if Ni∈ C < 1/u(Ni) > if Ni∈ LN (4.1)
where Reduction(hπi(Xj)/L(Xj, δ)Xj∈Succ(Ni)i) is defined by the following equation as we
have seen in Chapter 2:
Reduction(hλ1/L1, . . . , λm/Lmi) = h max j=1..m(λj⊗ λ j 1)/u1, . . . , max j=1..m(λj⊗ λ j n)/uni (4.2)
⊗ is the product operator in the case of numerical possibility theory and the min operator in the case of its qualitative counterpart.
Since, the operators max, ∗ and min used in the reduction operation are polytime, Equation (4.2) defines a polytime computation of the reduced lottery.
Proposition 4.1 For any strategy δ in ∆, a simple possibilistic lottery reduction equivalent to δ can be computed in polytime.
Proof. [Proof of Proposition 4.1]
Let δ ∈ ∆ = {(D0, δ(D0)), . . . , (Di, δ(Di)), . . . , (Dn, δ(Dl))} be a complete and sound strategy.
We first compute the compound lottery corresponding to δ, merging each decision node Di in δ with the chance node in δ(Di), say Ciδ. We get a compound lottery L = {C0δ, . . . , Ciδ, . . . , Clδ} ; the merging is performed linearly in the number of decision nodes in the strategy.
Then we can suppose without loss of generality that the nodes are numbered in such a way that i < j implies that Ciδ does not belong to the subtree rooted Cjδ (we label the nodes from the root to the leaves).
Then, for i = m to 1, we replace each compound lottery
Ciδ = hpri(Xi1)/Xi1, . . . , pri(Xiki)/Xikii by its reduction, where Succ(C
δ i) = {Xi1, . . . , Xiki} is the set of successors of C
δ
iand ki = |Succ(Ciδ)|. Because we pro- ceed from the leaves to the root, the Xi1 are simple lotteries. Since the min and max operation are linear, the reduction of this 2 level compound lottery is linear in the size of the compound lottery. The size of the resulting compound lottery is bounded by the sum of the size of the elementary lotteries before reduction, and thus linear. In any case, it is bounded by the number of levels in the scale, which is itself bounded by the number of edges and leaves in the tree (for the case where all the possibility degrees and all the utility degrees are different). Hence a complexity of the reduction is bounded by O(|E + LN |), where E is the number of edges and LN is the number of leave nodes in the strategy.
Thanks to the backward recursion, each node in the strategy is visited only once. Thus a global complexity is bounded by O(l.(E + LN )), where l the number of chance nodes in the strategy.
We are now in position to compare strategies, and thus to define the notion of optimality. Let O be one of the possibilistic decision criteria defined in Chapter 2 (i.e. depending on the application, ≥O is either ≥LΠ, or ≥LN, or the order induced by Upes, or by Uopt, etc.). A strategy δ ∈ ∆, is said to be optimal w.r.t. ≥O iff:
∀δ0 ∈ ∆, Reduction(δ) ≥OReduction(δ0). (4.3)
Notice that this definition does not require the full transitivity (nor the completeness) of ≥O and is meaningful as soon as the strict part of ≥O or >O, is transitive. This means that it is applicable to the preference relations that rely on the comparison of global utilities (qualitative utilities, binary utility and Choquet integrals) but also to ≥LN and ≥LΠ. We show in the following that the complexity of the problem of optimization depends on the criterion at work.
Like probabilistic decision trees, the simplest solving method of possibilistic decision trees consists on an exhaustive enumeration of all possible strategies in the decision tree which will be compared w.r.t decision criterion. The following example illustrates this process using ChN.
Example 4.2 Let us evaluate the decision tree in Figure 4.1 using necessity-based Choquet integrals as a decision criterion in the context of qualitative possibility theory.
We can distinguish, in Table 4.1, 5 possible strategies (∆ = {δ1, δ2, δ3, δ4, δ5}) where Li is the lottery of the strategy δi:
δi Li ChN(Li) δ1= {(D0, C1), (D1, C3), (D2, C5)} h1/1, 0.3/4, 0/5i 1 δ2= {(D0, C1), (D1, C3), (D2, C6)} h1/1, 0.5/2, 0.5/5i 1 δ3= {(D0, C1), (D1, C4), (D2, C5)} h0.2/0, 0.5/1, 1/4i 2.3 δ4= {(D0, C1), (D1, C4), (D2, C6)} h0.2/0, 0.5/2, 1/4, 0.5/5i 2.6 δ5 = {(D0, C2)} h1/1, 0.7/2i 1.7
So, the optimal strategy in this decision tree is δ4 with ChN(δ4) = 2.6 as it is shown in Figure 4.2. D0 D2 D1 C1 C2 C3 C4 C5 C6 1 5 0 4 1 4 2 5 1 2 1 0 0.2 1 1 0.3 1 0.5 1 0.7 1 0.5
Figure 4.2: The optimal strategy δ∗= {(D0, C1), (D1, C4), (D2, C6)}
Finding optimal strategies in possibilistic decision trees via an exhaustive enumeration of ∆ is a highly computational task. For instance, in a possibilistic decision tree with n decision nodes and a branching factor equal to 2, the number of potential strategies is in O(2
√
n) (exactly like probabilistic decision trees since the two kinds of decision trees have the same graphical component). Based on the work of [39], we can propose the following result:
Proposition 4.2 In a possibilistic decision tree with n nodes and a branching factor equal to 2, the number of potential strategies is in O(2
√ n).
Proof. [Proof of Proposition 4.2]
Suppose that we have a binary decision tree such that we have 4i decision nodes in depth 2i (1 decision node in depth 0,. . . , 16 decision nodes in depth 4). We will proceed by backward induction to compute the number of strategies according to the depth in the
decision tree.
For decision nodes which have no decision nodes in its successors we distinguish 2 strategies. Then we proceed by recurrence and the number of strategies starting by a chance node is equal to the product of the numbers of strategies beginning from its children. For decision nodes, the number of strategies is equal to the sum of the number of strategies of its children.
The total number of strategies is equal to a sequence 2u2k−1 when k is the number of decision nodes in a path from a decision node to a utility node. The general term of this sequence is equal to 2(2k+1−1). So, the number of strategies in the decision tree pertains to O(2
√ n).
For standard probabilistic decision trees, where the goal is to maximize expected utility (EU), an optimal strategy can be computed in polytime (with respect to the size of the tree) via the dynamic programming which builds the best strategy backwards, optimizing the decisions from the leaves of the tree to its root (see Algorithm 4.1).
Regarding possibilistic decision trees, Garcia and Sabbadin [33] have shown that such a method can also be used to get a strategy maximizing Upes and Uopt. The reason is that like EU, these possibilistic decision criteria satisfy the key property of weak monotonicity stating that the combination of L (resp. L0) with L”, does not change the initial order induced by O between L and L0 - this allows dynamic programming to decide in favor of L or L0 before considering the compound decision.
Formally for any decision criterion O over possibilistic lotteries, ≥Ois said to be weakly monotonic iff whatever L, L0 and L”, whatever (α,β) such that max(α, β) = 1:
L OL0 ⇒ hα/L, β/L”i Ohα/L0, β/L”i. (4.4)
Given any preference order O (satisfying the weak monotonicity property) among possibilistic lotteries, the possibilistic counterpart of dynamic programming algorithm (Al- gorithm 3.1) is depicted by Algorithm 4.1. When each chance node is reached, an optimal sub-strategy is built for each of its children - these sub-strategies are combined w.r.t. their possibility degrees, and the resulting compound strategy is reduced: we get an equivalent simple lottery, representing the current optimal sub-strategy. When a decision node X is reached, a decision Y∗ leading to a sub-strategy optimal w.r.t O is selected among all the possible decisions Y ∈ Succ(X), by comparing the simple lotteries equivalent to each sub strategies.
This procedure crosses each edge in the tree only once. When the comparison of simple lotteries by O (Line (2)) and the reduction operation on a 2-level lottery (Line (1)) can be performed in polytime, its complexity is polynomial w.r.t the size of the tree as stated by the following Proposition:
Proposition 4.3 If Osatisfies the monotonicity property, then dynamic programing com- putes a strategy optimal w.r.t O in polynomial time with respect to the size of the decision tree.
Proof. [Proof of Proposition 4.3]
The principle of the Backward induction method at work in dynamic programming is to eliminate sub-strategies that are not better than the optimal sub-strategies. The principle of monotonicity writes:
L O L0 ⇒ hα/L, β/L”i O hα/L0, β/L”i.
It guarantees that the elimination of sub-strategies that are not strictly better than their concurrents is sound and complete for the decision trees of size 2. Notice that L O L0 does not imply that L0 does not belong to an optimal strategy but it implies that if L0 belongs to an optimal strategy, so does L. When, a unique strategy among the optimal one is searched for, the algorithm can forget about L0.
The sequel on the proof is direct, by recursion on the depth on the decision tree. Let us denote hα/L, β/L”i by L1 and hα/L0, β/L”i by L2. Indeed, from L ≥O L0 ⇒ L1 O L2, we get that L ≥O L0⇒ hγL1, δL3i O hγL2, δL3i and so on.
Algorithm 4.1: Dynamic programming Data: In: a node X, In/Out: a strategy δ Result: A lottery L
begin
for i ∈ {1, . . . , n} do L[ui] ← 0
if N ∈ LN then L[u(N )] ← 1 if N ∈ C then
% Reduce the compound lottery foreach Y ∈ Succ(N ) do
LY ← P rogDyn(Y, δ)
for i ∈ {1, . . . , n} do
L[ui] ← max(L[ui], (πN(Y ) ⊗ LY[ui])) (Line (1))
if N ∈ D then
% Choose the best decision Y∗← Succ(N ).f irst foreach Y ∈ Succ(N ) do LY ← P rogDyn(Y, δ) if LY >OLY∗ then Y∗← Y (Line (2)) δ(N ) ← Y∗ L ← LY∗ return L end
In Line 1 of Algorithm 4.1, ⊗ is the min operator in the case of qualitative possibility theory and the product operator in the case of numerical possibility theory. We will see in the following that, beyond Upes and Uopt criteria, several other criteria satisfy the mono- tonicity property and that their optimization can be managed in polytime by dynamic programming. The possibilistic Choquet integrals, on the contrary, do not satisfy weak monotonicity; we will show that they lead to NP-Complete decision problems.
Formally, for any of the possibilistic optimization criteria, the corresponding decision problem can be defined as follows:
Definition 4.1 [DT-OPT-O](Strategy optimization w.r.t. an optimization criterion O in possibilistic decision trees)
INSTANCE: A possibilistic Decision Tree T , a level α.
QUESTION: Does there exist a strategy δ ∈ ∆ such as Reduction(δ) ≥Oα?
For instance DT-OPT-ChN (resp. DT-OPT-ChΠ, DT-OPT-Upes, DT-OPT-Uopt, DT-OPT- P U , DT-OPT-LN , DT-OPT-LΠ and DT-OPT-OM EU ) corresponds to the optimization
of the possibilistic qualitative utility ChN (resp. ChΠ, Upes and Uopt, P U , LN , LΠ and OM EU ). Each one of these decision problems will be studied in what follows.